Document Type : Research Paper
Authors
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Shahrekord, P.O.Box 88186-34141, Shahrekord, Iran.
Abstract
In this paper, we define the concepts of modified intuitionistic probabilistic metric spaces, the property (E.A.) and the common property (E.A.) in modified intuitionistic probabilistic metric spaces.
Then, by the common
property (E.A.), we prove some common fixed point theorems in modified intuitionistic Menger probabilistic metric spaces satisfying an implicit relation.
Keywords
Main Subjects
[1] M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270 (2002), pp. 181-188.
[2] S.S. Ali, J. Jain, and A. Rajput, A Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces, Int. J. Sci. Eng. Res., 4 (2013), pp. 1-6.
[3] J. Ali, M. Imdad, D. Mihet, and M. Tanveer, Common fixed points of strict contractions in Menger spaces, Acta Math. Hungar., 132 (2011), pp. 367-386.
[4] S.S. Chang, Y.J. Cho, and S.M. Kang, Nonlinear operator theory in probabilistic metric spaces, Nova Science Publishers Inc., New York, 2001.
[5] G. Deschrijver and E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets Syst., 133 (2003), pp. 227-235.
[6] J.X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal., 71 (2009), pp. 1833-1843.
[7] J.X. Fang and Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Anal., 70 (2009), pp. 184-193.
[8] M. Goudarzi, A generalized fixed point theorem in intuitionistic Menger spaces and its application to integral equations, Int. J. Math. Anal., 5 (2011), pp. 65-80.
[9] H.R. Goudarzi and M. Hatami Saeedabadi, On the definition of intuitionistic probabilistic 2-metric spaces and some results, J. Nonlinear Anal. Appl., (2014), pp. 1-8.
[10] O. Hadzic and E. Pap, A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces, Fuzzy Sets Syst., 127 (2002), pp. 333-344.
[11] O. Hadzic, E. Pap, and M. Budincevic, A generalisation of tardiffs fixed point theorem in probabilistic metric spaces and applications to random equations, Fuzzy Sets Syst., 156 (2005), pp. 124-134.
[12] M. Imdad, J. Ali, and M. Hasan, Common fixed point theorems in modified intuitionistic fuzzy metric spaces, Iran. J. Fuzzy Syst., 5 (2012), pp. 77-92.
[13] G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9 (1986), pp. 771-779.
[14] I. Kubiaczyk and S. Sharma, Some common fixed point theorems in Menger space under strict contractive conditions, Southeast Asian Bull. Math., 32 (2008), pp. 117-124.
[15] S. Kutukcu, A. Tuna, and A.T. Yakut, Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations, Appl. Math. & Mech., 28 (2007), pp. 799-809.
[16] Y. Liu and Z. Li, Coincidence point theorem in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst., 158 (2007), pp. 58-70.
[17] S. Manro and Sumitra, Common New Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces Using Implicit Relation, Appl. Math., 4 (2013), pp. 27-31.
[18] S. Manro, S.S. Bhatia, and S. Kumar, Common fixed point theorem for weakly compatible maps satisfying E.A. property in intuitionistic Menger space, J. Curr. Eng. & Maths, 1 (2012), pp. 5-8.
[19] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA, 28 (1942), pp. 535-537.
[20] D. Mihet, Multivalued generalisations of probabilistic contractions, J. Math. Anal. Appl., 304 (2005), pp. 464-472.
[21] S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japonica, 36 (1991), pp. 283-289.
[22] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004), pp. 1039-1046.
[23] B. Schweizer and A. Sklar, Probabilistic metric spaces, P. N. 275, North-Holland Seri. Prob. & Appl. Math., North-Holland Publ. Co. New York, 1983.
[24] B. Schweizer and A. Sklar, Statistical metric spaces, Pacifc J. Math., 10 (1960), pp. 313-334.
[25] V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Sys. Theory, 6 (1972), pp. 97-102.
[26] A. Sharma, A. Jain, and S. Choudhari, Sub-compatibility and fixed point theorem in intuitionistic Menger space, Int. J. Theor. & Appl. Sci., 3 (2011), pp. 9-12.
[27] R. Shrivastava, A. Gupta, and R. N. Yadav, Common fixed point theorem in intuitionistic Menger space, Int. J. Math. Arch., 2 (2011), pp. 1622-1627.
[28] B. Singh and S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301 (2005), pp. 439-448.
[29] M. Tanveer, M. Imdad, D. Gopal, and D. Kumar Patel, Common fixed point theorems in modified intuitionistic fuzzy metric spaces with common property (E.A.), Fixed Point Theory Appl., pp. 1-12.
[30] S. Zhang, M. Goudarzi, R. Saadati, and S. M. Vaezpour, Intuitionistic Menger inner product spaces and applications to integral equations, Appl. Math. Mech. Engl. Ed., 31 (2010), pp. 415-424.
[2] S.S. Ali, J. Jain, and A. Rajput, A Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces, Int. J. Sci. Eng. Res., 4 (2013), pp. 1-6.
[3] J. Ali, M. Imdad, D. Mihet, and M. Tanveer, Common fixed points of strict contractions in Menger spaces, Acta Math. Hungar., 132 (2011), pp. 367-386.
[4] S.S. Chang, Y.J. Cho, and S.M. Kang, Nonlinear operator theory in probabilistic metric spaces, Nova Science Publishers Inc., New York, 2001.
[5] G. Deschrijver and E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets Syst., 133 (2003), pp. 227-235.
[6] J.X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal., 71 (2009), pp. 1833-1843.
[7] J.X. Fang and Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Anal., 70 (2009), pp. 184-193.
[8] M. Goudarzi, A generalized fixed point theorem in intuitionistic Menger spaces and its application to integral equations, Int. J. Math. Anal., 5 (2011), pp. 65-80.
[9] H.R. Goudarzi and M. Hatami Saeedabadi, On the definition of intuitionistic probabilistic 2-metric spaces and some results, J. Nonlinear Anal. Appl., (2014), pp. 1-8.
[10] O. Hadzic and E. Pap, A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces, Fuzzy Sets Syst., 127 (2002), pp. 333-344.
[11] O. Hadzic, E. Pap, and M. Budincevic, A generalisation of tardiffs fixed point theorem in probabilistic metric spaces and applications to random equations, Fuzzy Sets Syst., 156 (2005), pp. 124-134.
[12] M. Imdad, J. Ali, and M. Hasan, Common fixed point theorems in modified intuitionistic fuzzy metric spaces, Iran. J. Fuzzy Syst., 5 (2012), pp. 77-92.
[13] G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9 (1986), pp. 771-779.
[14] I. Kubiaczyk and S. Sharma, Some common fixed point theorems in Menger space under strict contractive conditions, Southeast Asian Bull. Math., 32 (2008), pp. 117-124.
[15] S. Kutukcu, A. Tuna, and A.T. Yakut, Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations, Appl. Math. & Mech., 28 (2007), pp. 799-809.
[16] Y. Liu and Z. Li, Coincidence point theorem in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst., 158 (2007), pp. 58-70.
[17] S. Manro and Sumitra, Common New Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces Using Implicit Relation, Appl. Math., 4 (2013), pp. 27-31.
[18] S. Manro, S.S. Bhatia, and S. Kumar, Common fixed point theorem for weakly compatible maps satisfying E.A. property in intuitionistic Menger space, J. Curr. Eng. & Maths, 1 (2012), pp. 5-8.
[19] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA, 28 (1942), pp. 535-537.
[20] D. Mihet, Multivalued generalisations of probabilistic contractions, J. Math. Anal. Appl., 304 (2005), pp. 464-472.
[21] S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japonica, 36 (1991), pp. 283-289.
[22] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004), pp. 1039-1046.
[23] B. Schweizer and A. Sklar, Probabilistic metric spaces, P. N. 275, North-Holland Seri. Prob. & Appl. Math., North-Holland Publ. Co. New York, 1983.
[24] B. Schweizer and A. Sklar, Statistical metric spaces, Pacifc J. Math., 10 (1960), pp. 313-334.
[25] V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Sys. Theory, 6 (1972), pp. 97-102.
[26] A. Sharma, A. Jain, and S. Choudhari, Sub-compatibility and fixed point theorem in intuitionistic Menger space, Int. J. Theor. & Appl. Sci., 3 (2011), pp. 9-12.
[27] R. Shrivastava, A. Gupta, and R. N. Yadav, Common fixed point theorem in intuitionistic Menger space, Int. J. Math. Arch., 2 (2011), pp. 1622-1627.
[28] B. Singh and S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301 (2005), pp. 439-448.
[29] M. Tanveer, M. Imdad, D. Gopal, and D. Kumar Patel, Common fixed point theorems in modified intuitionistic fuzzy metric spaces with common property (E.A.), Fixed Point Theory Appl., pp. 1-12.
[30] S. Zhang, M. Goudarzi, R. Saadati, and S. M. Vaezpour, Intuitionistic Menger inner product spaces and applications to integral equations, Appl. Math. Mech. Engl. Ed., 31 (2010), pp. 415-424.
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